Horizontal Flows and Manifold Stochastics in Geometric Deep Learning

TL;DR

This paper presents novel stochastic methods for geometric deep learning on manifolds, enabling continuous filter transport and efficient convolution evaluation through manifold-valued sampling, grounded in differential geometry and statistics.

Contribution

It introduces two new constructions for geometric deep learning that incorporate stochastic processes on manifolds, advancing the theoretical foundation and practical implementation.

Findings

Transported convolutional filters that account for holonomy effects.

Efficient evaluation of manifold convolutions via sampling around diffusion means.

Theoretical analysis linking stochastic methods to existing geometric deep learning approaches.

Abstract

We introduce two constructions in geometric deep learning for 1) transporting orientation-dependent convolutional filters over a manifold in a continuous way and thereby defining a convolution operator that naturally incorporates the rotational effect of holonomy; and 2) allowing efficient evaluation of manifold convolution layers by sampling manifold valued random variables that center around a weighted diffusion mean. Both methods are inspired by stochastics on manifolds and geometric statistics, and provide examples of how stochastic methods -- here horizontal frame bundle flows and non-linear bridge sampling schemes, can be used in geometric deep learning. We outline the theoretical foundation of the two methods, discuss their relation to Euclidean deep networks and existing methodology in geometric deep learning, and establish important properties of the proposed constructions.